a) The vector parametric equation for the helical path is r(t) = <28cos(12πt), 28sin(12πt), 5t*12>, where t is the parameter and 0 ≤ t ≤ 1.
b) You walked a total distance of 840π feet.
a) To find the vector parametric equation for the helical path, we first note that the spiral staircase makes 6 complete revolutions as we climb from the bottom to the top. This means that as we climb 60 feet, we will make 6 full revolutions around the z-axis. Since the radius of the spiral is 28 feet, we can use the equation for a helix to describe our path.
r(t) = <28cos(θ), 28sin(θ), ht>
where θ is the angle of rotation around the z-axis and h is the height we have climbed. We know that we make 6 full revolutions, or 12π radians, as we climb from the bottom to the top, so we can substitute θ = 12πt. We also know that we climb 60 feet in one hour, so our height function is h(t) = 5t*12.
Thus, our vector parametric equation for the helical path is r(t) = <28cos(12πt), 28sin(12πt), 5t*12>, where t is the parameter and 0 ≤ t ≤ 1.
b) To find the distance we walked, we need to integrate the magnitude of the velocity vector over the interval [0,1].
The velocity vector is the derivative of the position vector:
v(t) = < -336πsin(12πt), 336πcos(12πt), 60>
The magnitude of the velocity vector is:
|v(t)| = sqrt((-336πsin(12πt))^2 + (336πcos(12πt))^2 + 60^2) = 336π
Thus, the distance we walked is:
distance = ∫|v(t)|dt from 0 to 1 = ∫336π dt from 0 to 1 = 336π(1-0) = 336π
Therefore, we walked a total distance of 840π feet.
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T
2
√
k
aº
90°
Special Right Triangles
2
If the diagonal of a square produces an isosceles right
triangle, then you should be able to find the measures
of the base angles.
What is the measure of angles a and b?
angle
a
b
measure
check
X
X
The base angles a and b have their measures to be 45 degrees each
Calculating the base angles a and b?If the diagonal of a square produces an isosceles right triangle, then the two legs of the right triangle are congruent, meaning the two base angles are congruent as well.
Since the triangle is a right triangle, one of the base angles must be 45 degrees (which is half of the right angle).
Therefore, both base angles are 45 degrees.
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whats the answer for log b + z log c in condense form
Answer:
log b.c^z
Step-by-step explanation:
from the theorem a log b=log b^a
z log c= log c^z
from the theorem log a + log b= log ab
log b+ log c^z= logb. c^z will be the final answer
What is the most descriptive name for the illustration below?
A. quadrilateral
B. polygon
C. rhombus
D. trapezoid
The most descriptive name for the illustration below is C. rhombus.
Define term Rhombus?Because it has four sides, a rhombus is a geometric shape that is a quadrilateral. Because its opposite sides are parallel and their angles are equal, it is a unique kind of parallelogram.
The most descriptive name for the illustration above is C. rhombus.
A rhombus is a type of quadrilateral with four sides of equal length and opposite angles that are equal to each other. It can also be defined as a parallelogram with equal diagonals. The illustration appears to have all sides of equal length and opposite angles that are equal, thus making it a rhombus.
While the other options, such as quadrilateral, polygon, and trapezoid, also apply to the illustration, they are less specific than the term rhombus.
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Felicia is deciding on her schedule for next semester. She must take each of the following classes: English 102, Spanish 102, History 102, and College Algebra. If there are 15 sections of English 102, 9 sections of Spanish 102, 12 sections of History 102, and 13 sections of College Algebra, how many different possible schedules are there for Felicia to choose from? Assume there are no time conflicts between the different classes.
Okay, here are the steps to solve this problem:
* There are 15 sections of English 102 to choose from
* There are 9 sections of Spanish 102 to choose from
* There are 12 sections of History 102 to choose from
* There are 13 sections of College Algebra to choose from
So for English 102 Felicia has 15 options, for Spanish 102 she has 9 options, for History 102 she has 12 options, and for College Algebra she has 13 options.
By the Fundamental Principle of Counting, the total number of possible schedules is:
15 * 9 * 12 * 13 = 16,920
Therefore, the total number of possible schedules for Felicia is 16,920
find the area of the shaded region round to the nearest hundredth where necessary 23.8 21 15
Determine the equation of straight line passing through the point (1,0) and (2, - 3)
first you need to find the gradient of the line and to do that you divide the height of a part of the line by its length. so to find the height we do -3-0=-3 and to find the length we do 2-1=1. then divide -3 by 1 and the gradient is -3. so now we know the equation is y=-3x+b. to find b we replace y with the y coordinate of one of the 2 points. lets take the first point. so it will be 0=-3+b. Now we can easily find b by doing 0-(-3) which is 3. so the equation of the line is y=-3x+3. Also please mark this as brainliest answer if you found it helpful thanks.
Answer:
y = - 3x + 3
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (1, 0 ) and (x₂, y₂ ) = (2, - 3 )
m = [tex]\frac{-3-0}{2-1}[/tex] = [tex]\frac{-3}{1}[/tex] = - 3 , then
y = - 3x + c ← is the partial equation
to find c substitute either of the 2 points into the partial equation
using (2, - 3 )
- 3 = - 3(2) + c = - 6 + c ( add 6 to both sides )
3 = c
y = - 3x + 3 ← equation of line
1. (i) Find the first 3 terms in the expansion of (2 - y)^5 in ascending powers of y.
(ii) Use the result in part (i) to find the coefficient of x² in the expansion of (2 − (2x - x²))^5.
1) The first three terms in the expansion of [tex](2 - y)^{5}[/tex] in ascending powers of y are as follows:
32, -80y, and [tex]80y^2[/tex]
2)The [tex]x^{2}[/tex] coefficient in the expansion of (2 (2x - [tex]x^{2}[/tex]))5 is 80.
What is the binomial theorem?As the power increases, the expansion gets more difficult to compute. The Binomial Theorem can be used to easily calculate a binomial statement that has been raised to very big power.
(i) Using the binomial theorem, we can expand [tex](2 - y)^{5}[/tex] as follows:
[tex](2 - y)^{5}[/tex] = [tex]2^{5} - 5(24)y^{1} + 10(23)y^{2} + 10(23)y^{3} + 5(2)y^{4} - y^{5}[/tex]
By combining the powers of two and simplifying, we get:
[tex]32 - 80y + 80y^{2} - 40y^3 + 10y^4 - y^5[/tex]
As a result, the first three terms in the expansion of [tex](2 - y)^{5}[/tex] in ascending powers of y are as follows:
32, -80y, and [tex]80y^2[/tex]
(ii) Using the result of section (i), we can expand [tex](2 - (2x - x^{2} ))^{5}[/tex] by substituting y with 2x - x2:
[tex](2 - (2x - x^2))^5 = 2^5 - 5(2^4)(2x - x^2) + 10(2^3)(2x - x^2)^2 - 10(2^2)(2x - x^2)^3 + 5(2)(2x - x^2)^4 - (2x - x^2)^5[/tex]
By combining the powers of two and simplifying, we get:
[tex]32 - 80x + 80x^2 - 40x^3 + 10x^4 - x^5[/tex]
As a result, the [tex]x^{2}[/tex] coefficient in the expansion of (2 (2x - [tex]x^{2}[/tex]))5 is 80.
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Please see the attached
a. Monthly payment for the bank's car loan is $407.67
b. Monthly payment for the savings and loan association's car loan is $315.99
c. Total amount paid would be $2,035 less for the bank's car loan than for the savings and loan association's car loan.
What is interest rate?The cost of borrowing money, usually expressed as a percentage of the amount borrowed, is what a lender charges a borrower to use their money. This cost is known as an interest rate.
(a) To find the monthly payment for the bank's car loan, we can use the formula for the present value of an annuity:
[tex]PV = PMT * \frac{1 - (1 + \frac{r}{n})^{(-n*t)}}{\frac{r}{n} }[/tex]
putting the given values,
⇒ [tex]21000 = PMT * \frac{1 - (1 + \frac{0.065}{12})^{(-12*5)}}{\frac{0.065}{12} }[/tex]
Solving for PMT, we get:
PMT = $407.67
Therefore, the monthly payment for the bank's car loan is $407.67
(b) To find the monthly payment for the savings and loan association's car loan, we can use the same above formula:
where PV is still $21,000, PMT is the monthly payment, r is still 0.065, n is still 12, but t is now 7 years x 12 months/year = 84 payments.
putting the given values,
[tex]21000 = PMT * \frac{1 - (1 + \frac{0.065}{12})^{(-12*7)}}{\frac{0.065}{12} }[/tex]
Solving for PMT, we get:
PMT = $315.99
Therefore, the monthly payment for the savings and loan association's car loan is $315.99.
(c) Bank's car loan: $407.67 x 60 = $24,460.20
Savings and loan association's car loan: $315.99 x 84 = $26,495.16
Therefore, the bank's car loan would have the lowest total amount to pay off, by: $26,495.16 - $24,460.20 = $2,034.96
Therefore, the total amount paid would be $2,035 less for the bank's car loan than for the savings and loan association's car loan.
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If the first 4 terms of an infinite geometric sequence are 150, 120, 96, and 76.8, then the sum of all the terms in the sequence is ______.
Answer:
The common ratio of this sequence is 120/150 = 4/5, so we have:
[tex] \frac{150}{1 - \frac{4}{5} } = \frac{150}{ \frac{1}{5} } = 150 \times 5 = 750[/tex]
So the sum of all the terms in this sequence is 750.
Prepare a statement of cash flows for Business Solutions using the indirect method for the three months ended March 31, 2022. Owner Santana Rey contributed $25,000 to the business in exchange for additional stock in the first quarter of 2022 and has received $4,800 in cash dividends.
Giving a test to a group of students, the grades and gender are summarized below
A B C Total
Male 12 15 4 31
Female 9 8 16 33
Total 21 23 20 64
If one student is chosen at random, find the probability that the student was male GIVEN they got a 'C':
Answer: [tex]\frac{1}{5}[/tex]
Step-by-step explanation:
Males who got a C = 4
Total amount of people who got a C = 20
[tex]\frac{4}{20}[/tex]
[tex]\frac{4}{20}[/tex] simplifies to [tex]\frac{1}{5}[/tex]
Can someone please help me
The height of point A is 0 feet. However, this is not the actual height of point A, as it is at the highest point on the Ferris Wheel, which is 185 feet as calculated previously.
How to solveThe height of point A on the Ferris Wheel can be calculated using the sine function, as it is the vertical component of the point's position. The center of the Ferris Wheel is 100 feet off the ground, and its radius is 85 feet.
Let's denote the height of point A as h_A, which is the vertical distance from the center of the Ferris Wheel to point A.
Since point A is at the 0 radian position, it is at the highest point on the Ferris Wheel. At this point, the height of point A is equal to the sum of the radius and the center's height:
h_A = radius + center's height
h_A = 85 + 100
h_A = 185 feet
So, the height of point A off the ground is 185 feet.
To estimate the value of h_A using the sine function, we can use the fact that at any given angle θ on the unit circle, the height of the point on the circle is given by the sine of that angle multiplied by the radius. In this case, since point A is at the highest point on the Ferris Wheel, we can use the sine of 0 radians (which is 0) to estimate the height of point A:
h_A = radius * sin(0)
h_A = 85 * 0
h_A = 0
So, using the sine function, we estimate that the height of point A is 0 feet. However, this is not the actual height of point A, as it is at the highest point on the Ferris Wheel, which is 185 feet as calculated previously.
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What is the length of side b? (Hint: There are 3 sides and 1 angle)
The length of side b is given as follows:
b = 2.86 m.
What is the law of sines?Suppose we have a triangle in which:
Side with a length of a is opposite to angle A.Side with a length of b is opposite to angle B.Side with a length of c is opposite to angle C.The lengths and the sine of the angles are related as follows:
[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]
The sum of the measures of the internal angles of a triangle is of 180º, hence the missing angle is obtained as follows:
B + 30 + 80 = 180
B = 70º.
Then the relation, by the law of sines, is given as follows:
sin(70º)/b = sin(80º)/3
Meaning that the length of side b is given as follows:
b = 3 x sine of 70 degrees/sine of 80 degrees
b = 2.86 m.
Missing InformationThe triangle is given by the image presented at the end of the answer.
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A recent survey reported that 87% of Americans approve of human embryonic stem cell research. If an American is selected at random, find the probability that he or she will disapprove or have no opinion on the issue. The probability that he or she will disapprove or have no opinion on the issue is %?
The probability that an American will disapprove or have no opinion on the issue is 13%.
What is Probability?Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. Probability is used to describe random phenomena, such as the roll of a dice, the flip of a coin, the occurrence of an earthquake, or the outcome of a hand of poker. It is also used to describe the likelihood of certain events happening based on past data or current conditions.
This can be calculated by subtracting the reported approval rate of 87% from 100%. This means that 13% of Americans disapprove or have no opinion on the issue of human embryonic stem cell research.
It is important to note that this survey's data only pertains to the opinions of Americans and not to any other group of people. Therefore, the probability that any individual randomly selected from the global population will disapprove or have no opinion on the issue may be different.
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75% of flights arriving in Memphis are on time if the FAA Selects 60 random flights what is the probability that more than 80% of a simple flights are on time?
Suppose you physically simulate the random process of rolling a single die. (a) After 1010 rolls of the die, you observe a "one" 44 times. What proportion of the rolls resulted in a "one"? (b) After 2020 rolls of the die, you observe a "one" 22 times. What proportion of the rolls resulted in a "one"?
Answer:
(a) 4.356% (b) 1.089%
Step-by-step explanation:
The rolling of a single die is an example of a discrete uniform distribution, where each of the six possible outcomes has an equal probability of 1/6.
(a) After 1010 rolls of the die, observing a "one" 44 times, the proportion of rolls resulting in a "one" is:
Proportion = Number of "ones" / Total number of rolls
= 44 / 1010
= 0.04356
Therefore, approximately 4.356% of the rolls resulted in a "one".
(b) After 2020 rolls of the die, observing a "one" 22 times, the proportion of rolls resulting in a "one" is:
Proportion = Number of "ones" / Total number of rolls
= 22 / 2020
= 0.01089
Therefore, approximately 1.089% of the rolls resulted in a "one".
Answer:
A) 40%
B) 10%
Step-by-step explanation:
A) p = 4/10 = 40%
B) p = 2/20 = 10%
Kleen Company acquired patent rights on January 10 of Year 1 for $881,100. The patent has a useful life equal to its legal life of eight years. On January 7 of Year 4, Kleen successfully defended the patent in a lawsuit at a cost of $49,200.
Required:
a. Determine the patent amortization expense for Year 4 ended December 31.
b. Journalize the adjusting entry on December 31 of Year 4 to recognize the amortization. Refer to the Chart of Accounts for exact wording of account titles.
Cruz has 64 stamps, of which 1/8 of the stamps have bees on them, and 3/16 of the stamps have flowers. The rest of the stamps have butterflies. How many stamps have butterflies on them? Explain.
Answer:
44 butterfly stamps
Step-by-step explanation:
1/8=2/16
2/16+3/16=5/16
16/16-5/16=11/16
64÷16=4
4×11=44
44/64
Need help with his page 50 points
Answer: B, C, B
Step-by-step explanation:
see images for explanation
Answer:
[tex]\textsf{11)} \quad \text{b.}\;\;2\frac{3}{4}[/tex]
[tex]\textsf{12)} \quad \text{c.}\;\;7\frac{1}{2}[/tex]
[tex]\textsf{13)} \quad \text{b.}\;\;\dfrac{2}{3}x+4=10[/tex]
Step-by-step explanation:
Question 11To find the answer to the given equation when y = 4, substitute y = 4 into the equation:
[tex]\begin{aligned} y=4 \implies \dfrac{3}{4}+\dfrac{2(4)}{4}&= \dfrac{3}{4}+\dfrac{8}{4}\\\\&= \dfrac{3}{4}+2\\\\&= 2\frac{3}{4}\end{aligned}[/tex]
Therefore, the answer is 2³/₄.
[tex]\hrulefill[/tex]
Question 12To find the answer to the given equation when y = 8, substitute y = 8 into the equation:
[tex]\begin{aligned}y=8 \implies \dfrac{12}{8}+\dfrac{3(8)}{4}&= \dfrac{12}{8}+\dfrac{24}{4}\\\\&= \dfrac{12}{8}+6\\\\&= \dfrac{8+4}{8}+6\\\\&= \dfrac{8}{8}+\dfrac{4}{8}+6\\\\&= 1+\dfrac{1}{2}+6\\\\&=7\frac{1}{2}\end{aligned}[/tex]
Therefore, the answer is 7¹/₂.
[tex]\hrulefill[/tex]
Question 13To determine which equation is true if x = 9, substitute x = 9 into each equation.
[tex]\begin{aligned} \text{a)} \quad \dfrac{2}{3}(9)-6&=12\\\\\dfrac{18}{3}-6&=12\\\\6-6&=12\\\\0&=12\end{aligned}[/tex]
[tex]\begin{aligned} \text{b)} \quad \dfrac{2}{3}(9)+4&=10\\\\\dfrac{18}{3}+4&=10\\\\6+4&=10\\\\10&=10\end{aligned}[/tex]
[tex]\begin{aligned} \text{c)} \quad 3(9)-12&=21\\27-12&=21\\15&=21\end{aligned}[/tex]
[tex]\begin{aligned} \text{d)} \quad 3(9)+12&=19\\27+12&=19\\39&=19\end{aligned}[/tex]
Therefore, the only equation that holds true is option b.
ASAPP Rajindri, a physician assistant who works in an emergency room, earns $163 for every two hours that she works.
Which equation represents the relationship between d, the number of dollars Rajindri earns, and t, the amount of time Rajindri works, in hours?
A. d= 163 + t
B. d= 163/2 × t/2
C. d = 163t
D. d = 81.50t
Answer:
C. d = 163t
Step-by-step explanation:
This equation represents a direct proportionality between the number of dollars earned (d) and the amount of time worked (t), with a constant of proportionality of 163 dollars per hour.
Hope this helps!
Allen plans to weld peices of metal together to create a metal cube the length of of each side of the cube will be 14 inches how much metal will he need to create the cube
Does each of these graphs have at least one Hamiltonian circuit? If so, find one.
The first shape does NOT have a Hamilton circuit
The second shape has a Hamilton circuit
The third shape does NOT have a Hamilton circuit
What is a Hamiltonian circuit?In mathematical graph theory, a Hamiltonian circuit- named after its originator William Rowan Hamilton, an acclaimed Irish mathematician of the 19th century– is a path within a graph that covers every vertex only once and begins and ends at the same vertex.
To put it differently, it's an enclosed movement that goes through each point in the graph exactly once. When the graph consists of a Hamiltonian circuit, this indicates to be a Hamiltonian graph. One can see many practical uses of Hamiltonian circuits such as computer science where they play a crucial role in algorithm analysis and design.
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You want to buy a $12,000 car. The company is offering a 2% interest rate for 48 months (4 years). What
will your monthly payments be?
Answer:
240
Step-by-step explanation:
factor each completely
6r2 + 31r + 18
Therefore, the expression 6r² + 31r + 18 factors completely into (3r + 2)(2r + 9).
What is factor?In mathematics, factoring is the process of finding the factors or divisors of a given number or algebraic expression. When we factor a number or expression, we find the numbers or expressions that can be multiplied together to obtain the original number or expression. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, since these numbers can be multiplied together in different combinations to obtain 12. Factoring is an important skill in algebra, as it helps us to simplify algebraic expressions and solve equations.
Here,
To factor the quadratic expression completely, we need to find two binomials whose product is equal to the given expression. We can start by looking for two numbers whose product is 6 x 18 = 108 and whose sum is 31. After some trial and error, we find that the numbers are 6 and 18.
We can now write the expression as:
=6r² + 31r + 18
18*6=108
27*4=108
27+4=31
= 6r² + 27r + 4r + 18
= 3r(2r + 9) + 2(2r + 9)
= (3r + 2)(2r + 9)
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please help me!!!!!!
Answer:2nd one
Step-by-step explanation:
Mathematics
Calculate the percentage, show your solution.
90% of 60
help, thanks!
Answer:
Your answer is 90% of 60 = 54.
When Caroline runs the 400 meter dash, her finishing times are normally distributed with a mean of 68 seconds and a standard deviation of 1.5 seconds. Using the empirical rule, determine the interval of times that represents the middle 99.7% of her finishing times in the 400 meter race.
The interval of times that represents the middle 99.7% of Caroline's finishing times is (63.5, 72.5) seconds
How to calculate the intervalIn this case, we are looking for the interval of times that represents the middle 99.7% of Caroline's finishing times.
The middle 99.7% of the data corresponds to three standard deviations from the mean in both directions
Three standard deviations from the mean is 3 x 1.5 = 4.5 seconds.
Therefore, the interval of times that represents the middle 99.7% of Caroline's finishing times is (68 - 4.5, 68 + 4.5) = (63.5, 72.5) seconds
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Elnaz rewrote the equation x² - 6x - 18 = 0 so that she can solve by completing the square. Which equation did she write?
A. (x-3)² = -9
B. (x-3)² = 18
C. (x-3)² = 27
D. (x-3)² = 54
Answer:
C
Step-by-step explanation:
(x-3)(x-3)=27
x^2-3x-3x+9=27
x^2-6x+9=27
-27. -27
x^2-6x-18=0
Answer:
(x-3)²=27
Step-by-step explanation:
x²+6x-18=0
add 18 to both sides of the equation
To complete the square add 3² to both sides and factor what is on the left side. You will be getting
(x-3)² = 27 is the answer to this problem
Answer:
C
Step-by-step explanation:
(x-3)(x-3)=27
x^2-3x-3x+9=27
x^2-6x+9=27
-27. -27
x^2-6x-18=0
Answer:
(x-3)²=27
Step-by-step explanation:
x²+6x-18=0
add 18 to both sides of the equation
To complete the square add 3² to both sides and factor what is on the left side. You will be getting
(x-3)² = 27 is the answer to this problem
Kenji walks 44 feet in 10 seconds. At this rate, how many miles does Kenji walk in an hour? Show your work. (1 mile= 5,280 feet)
Answer:
44 feet in 10 seconds = 44 x 6 (since there are 60 seconds in a minute) = 264 feet per minute
264 feet per minute x 60 minutes (since there are 60 minutes in one hour) = 15,840 feet per hour
15,840 feet per hour / 5,280 feet (since there are 5,280 feet in 1 mile) = 3 miles per hour
The lateral area of a cone is 473pi cm^2. The radius is 43 cm. Find the height to the nearest tenth
The slant height of the cone is about 11 cm. Using the Pythagorean theorem, the height is approximately 41.4 cm to the nearest tenth.
Use the formula for the lateral area of a cone: L = πrs, where L is the lateral area, r is the radius of the base, and s is the slant height.
Plug in the given values: L = 473π cm² and r = 43 cm. Then solve for s
L = πrs
473π = π(43)s
s = 473/43 ≈ 11
So the slant height of the cone is approximately 11 cm.
Use the Pythagorean theorem to find the height of the cone. Let h be the height of the cone. Then the Pythagorean theorem gives
h² + s² = r²
Substituting in the values for s and r, we get
h² + 11² = 43²
Simplifying this equation, we get
h² = 43² - 11²
Evaluating the right-hand side, we get
h² = 1764
Taking the square root of both sides, we get
h ≈ 41.4
So the height of the cone is approximately 41.4 cm to the nearest tenth.
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